The determination of the frequency components of a signal is important in many applications. For example, in an inverter which inverts DC power into AC power, it is typical practice to sense the frequency components of the inverter output in order to operate the inverter so as to control those frequency components. Specifically, it is typical practice to control the first harmonic, i.e. that frequency component having a frequency equal to the fundamental frequency of the inverter output, at a predetermined magnitude and phase, and to control as many of the other harmonic components as is practical so that they are suppressed in the inverter output.
An inverter is often used in a power conversion system, such as a variable speed, constant frequency (VSCF) power generating system. In a variable speed, constant frequency power generating system, a generator, typically a brushless, three-phase synchronous generator, is operated in a generating mode to convert variable speed motive power supplied by a prime mover into variable frequency AC power. The prime mover may be, for example, a gas turbine engine of an aircraft. The variable frequency AC power produced by the generator is rectified and provided as a DC signal over a DC link to an inverter.
The inverter inverts the DC signal on the DC link into a constant frequency AC inverter output for supply over a load bus to one or more AC loads. The inverter is controlled so that its constant frequency AC output has a desired fundamental frequency. However, the inverter output normally also includes a plurality of harmonics of the fundamental frequency; that is, each such harmonic has a frequency which is an integer multiple of the inverter output fundamental frequency.
Because such harmonics in an inverter output are, generally, undesirable, the inverter is normally controlled so that harmonics of the inverter output fundamental frequency are suppressed or eliminated. In order to control an inverter so as to suppress or eliminate these harmonics, the inverter output is analyzed, normally by a Fourier analysis, in order to determine the harmonic content therein, and the inverter is controlled in response to that harmonic content.
That is, a periodic signal can be represented by an infinite series of trigonometric terms according to the following equation: EQU f(t)=a.sub.o +a.sub.n cos (n.omega.t)+b.sub.n sin (n.omega.t)(1)
where f(t) is the periodic signal having a fundamental frequency f, a.sub.o is the average, i.e. DC, value of the periodic signal f(t), a.sub.n is the magnitude of the corresponding cosine component cos(n.omega.t), b.sub.n is the magnitude of the corresponding sine component sin(n.omega.t), n=1, 2, 3, 4, 5, . . . and is the harmonic number specifying each of the frequencies in the periodic signal f(t) (the fundamental frequency is considered to be the first harmonic, i.e. n=1), and .omega. (i.e., 2.pi.f) is the fundamental angular frequency at the fundamental frequency f. The values for a.sub.n and b.sub.n in equation (1) are typically determined by a Fourier analysis. This analysis involves multiplying the signal f(t) by a cosine function (i.e., cos(n.omega.t)) having a frequency determined by the harmonic number n and integrating the result over one period of the signal f(t) to determine a.sub.n. Similarly, b.sub.n in equation (1) is determined by multiplying the signal f(t) by a sine function (i.e., sin(n.omega.t)) having a frequency determined by the harmonic number n and integrating the result over one period of the signal f(t). By setting n=1, 2, 3, 4, 5 . . . , the values of a.sub.n and b.sub.n at the fundamental frequency and each of its harmonics can thus be determined.
Each harmonic can be further specified, if desired, by determining its magnitude and phase. The magnitude of a harmonic is simply the square root of the sum of the squares of its corresponding a and b values. Thus, the magnitude of harmonic n can be determined according to the following equation: ##EQU1## where mag.sub.n is the magnitude of harmonic n, a.sub.n is the a value in equation (1) for the harmonic n, and b.sub.n is the b value in equation (1) for the harmonic n. The phase of that harmonic is determined by the arc tangent of its corresponding b value divided by its corresponding a value. Thus, the phase of the harmonic n can be determined by the following equation: ##EQU2## where pha.sub.n is the phase of harmonic n, a.sub.n is the a value in equation (1) for the harmonic n, and b.sub.n is the b value in equation (1) for the harmonic n.
The approach of integrating f(t)sin(n.omega.t) and f(t)cos(n.omega.t), however, involves a great number of calculations and, therefore, requires a substantial amount of processing time to implement. A faster Fourier analysis can be made by performing a Discrete Fourier Transform, and its even faster form, the Fast Fourier Transform. In performing a Discrete Fourier Transform, a signal is sampled (i.e., tested for magnitude) at a sampling frequency. The samples are then used to determine the set of Fourier coefficients which define the fundamental and harmonic components of the signal being analyzed. In order to avoid aliasing error, the sampling frequency, i.e. the frequency at which the samples are taken, must be greater than the highest frequency of the harmonic components to be determined. This aliasing error generally increases as the sampling frequency decreases toward the frequency of the harmonic component to be determined. For example, if the sampling frequency is exactly equal to the frequency of the harmonic component to be determined, that harmonic component appears as a DC signal since it is being sampled at exactly the same phase in each of its cycles, and the aliasing error is consequently very large. Normally, the sampling frequency is at least twice the frequency of the harmonic component to be determined, but it is usually much higher. Because the sampling frequency needs to be large compared to the harmonic of interest in order to avoid aliasing errors, the amount of processing time required to determine the harmonics (including the fundament frequency and other harmonics) in the analyzed signal is consequently large.